Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

The existing literature gives efficient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend. Communicated by H. de Fraysseix and and J. Kratochv́ıl: submitted February 2000; revised May 2001. M. Kaufmann and R. Wiese, Few Bends, JGAA, 6(1) 115–129 (2002) 116